Affiliations: School of Aerospace Engineering and Applied Mechanics,
Tongji University, Shanghai, China
Note: [] Corresponding author: A.M. Yu, School of Aerospace Engineering
and Applied Mechanics, Tongji University, Shanghai, China. Tel.: +86 21
65983267; Fax: +86 21 65983267; E-mail: aimin.yu@163.com
Abstract: Free vibration equations for non-cylindrical (conical, barrel, and
hyperboloidal types) helical springs with noncircular cross-sections, which
consist of 14 first-order ordinary differential equations with variable
coefficients, are theoretically derived using spatially curved beam theory. In
the formulation, the warping effect upon natural frequencies and vibrating mode
shapes is first studied in addition to including the rotary inertia, the shear
and axial deformation influences. The natural frequencies of the springs are
determined by the use of improved Riccati transfer matrix method. The element
transfer matrix used in the solution is calculated using the Scaling and
Squaring method and Pad'e approximations. Three examples are presented for
three types of springs with different cross-sectional shapes under
clamped-clamped boundary condition. The accuracy of the proposed method has
been compared with the FEM results using three-dimensional solid elements
(Solid 45) in ANSYS code. Numerical results reveal that the warping effect is
more pronounced in the case of non-cylindrical helical springs than that of
cylindrical helical springs, which should be taken into consideration in the
free vibration analysis of such springs.
